Question

There are two radioactive substances A and B. Decay constant of B is two times that of A. Initially both have equal number of nuclei. After n half lives of A, rate of disintegration of both are equal. The value of n is-
(A). 4
(B). 2
(C). 1
(D). 5

Answer 1

Hint: Two radioactive substances have decay constants in the ratio 1:2. Therefore, their rates of disintegration will be in the same ratio. Therefore, the time taken by one substance to reduce to half its value is half the time taken by the other. Therefore their final rates of disintegration will depend on the number of nuclei remaining after their half lives.

Formula used:
$r=\lambda N$

Complete answer:
The probability of number of decays per unit time is known as decay constant. It is denoted by $\lambda $.
The half life of a radioactive substance is the time taken by it to disintegrate to half the amount. It is denoted by ${{T}_{1/2}}$.
The rate of disintegration is defined as the number of disintegrations taking place per second.
Let the decay constant of A be $\lambda $, then the decay constant of B will be $2\lambda $.
Given, initially, A and B have equal numbers of nuclei, let it be ${{N}_{0}}$.
The rate of disintegration is given by-
$r=\lambda N$
From the above equation, the rate of disintegration of A will be-
${{r}_{a}}=\lambda {{N}_{0}}$ - (1)
The rate of disintegration of B will be-
${{r}_{b}}=2\lambda {{N}_{0}}$ - (3)
Since the rate of disintegration of B is two times that of A, the time taken by B to reduce to half its value will be half that of A. Let the half life of A be $T$, then the half life of B will be $\dfrac{T}{2}$.
After one half life of A, its rate of disintegration will be-
${{r}_{a}}=\dfrac{\lambda {{N}_{0}}}{2}$
In one half life of A, B completes two of its half lives and reduces to $\dfrac{{{N}_{0}}}{4}$. Therefore, its rate of integration after one half life of A will be-
$\begin{align}
  & {{r}_{b}}=\dfrac{2\lambda {{N}_{0}}}{4} \\
 & \Rightarrow {{r}_{b}}=\dfrac{\lambda {{N}_{0}}}{2} \\
\end{align}$
Therefore after one half life of A, the rate of disintegration of both A and B is equal.
Therefore, the number of half lives after which the rate of disintegration of both A and B are equal is one.

Hence, the correct option is (C).

Note:
The number of nuclei decreases exponentially with time. The half life of a radioactive substance depends on the number of nuclei and the decay constant. The mean time period of the life of a radioactive substance is known as mean lifetime. During decay, a radioactive substance emits alpha, beta and gamma rays.